Nuclear power to the rescue (again! Lol) use excess heat from nuclear power generation and the water that gets steamed anyways for cooling and desalinate it. The brine problem can be solved by not desalinating endlessly but using desalination to create lakes and dump the brine in the desert where nuke waste is buried.
Nuclear power is, as always, overwhelmingly more espensive than the alternative, solar. We have a great deal of desert in California, with plenty of sunshine, and evaporating only when the sun is out is fine.
You wouldn't even need enclosed evaporation containers; let the wind carry the moisture to the mountains, raining into existing reservoirs.